The Federal Reserve Bank of Boston has published a public policy discussion document entitled, "Why Don’t Most Merchants Use Price Discounts to Steer Consumer Payment Choice?"
We find that steering consumers to debit and cash via price discounts reduces some merchants’ card costs. However, this cost reduction may be insufficient to offset the cost increase of administering price menus that vary by payment instrument. In addition, rewards buyers receive on credit card transactions may exceed the price discounts that merchants can provide. These factors may explain why steering via price discounts is not widely observed.
The full paper is here.